What is a Null Set and Why Does It Have Measure Zero?

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Discussion Overview

The discussion revolves around the concept of a null set, its definition, and its relationship to measure zero. Participants explore various interpretations and examples, including the empty set and countable subsets of real numbers, particularly rationals and irrationals.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant seeks clarification on the definition of a null set, expressing confusion about the term.
  • Another participant suggests that a null set generically refers to something that is zero.
  • It is proposed that the null set serves to distinguish between "something" and "nothing," with the null set representing nothing.
  • A counterpoint is raised that a null set could specifically refer to a set of measure zero.
  • Participants discuss the notation for the null set, with references to the symbols ø and the empty set.
  • There is a claim that the empty set is unique in having cardinality zero, but its relevance to the definition of a null set depends on the context.
  • A participant questions whether there are other sets with cardinality zero besides the empty set, expressing skepticism about the existence of such sets.
  • One participant provides an example involving the rationals as a null set, asking for clarification on why they are considered as such.
  • A mathematical explanation is offered, stating that countable subsets of real numbers, like the rationals, have measure zero through a specific construction involving intervals.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of a null set, with no consensus reached on its broader significance or the uniqueness of the empty set in this context.

Contextual Notes

The discussion highlights varying interpretations of the term "null set" and its relationship to measure zero, with some assumptions about context and definitions remaining unresolved.

mooberrymarz
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Wot exactly is a null set? i don't understand it. if anyone could explain that wuld be wikid. :cool:
 
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it would depend on the context, but it is, generically, something that is zero.
 
It exists to make difference between "something" and "nothing". A set that contains something is like something and the null set is nothing (contains nothing)
 
No, that is not necessarily true. A null set, could be, as I suspect it will be in this case, a set of measure zero.
 
ok..maybe my english is not good enough.. what is that called: ø?
 
isn't it a null set? or an empy set?
 
that is the empty set. It is a null set in the sense that its cardinality is zero.
 
is there any other set, whiches cardinality is zero exept the empty set?
I don't think so
 
No, it, the empty set, is unique. But that doesnt' have any bearing on what a null set is until we see what situation we are dealing with.
 
  • #10
For example.. the set of rationals and irrationals. rationals are a null set and wot bout irrationals?? Matt could u explain in dummy maths why rationals are a null set?
 
  • #11
As I thought, a null set is one that has measure zero. Example: and countable subset of R (Eg the rationals): let x_i be an enumeration of the set, round each point x_i consider the interval e/2^i, then the measure of the set is less than the sum over i of e^2^i = e. e was arbitrary hence it has measure zero.
 

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